Origin and Control of Dynamics of Hexagonal Patterns in a Photorefractive Feedback System

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1 Pergamon Chaos\ Solitons + Fractals Vol[ 09\ No[ 3Ð4\ pp[ 690Ð696\ 0888 Þ 0888 Elsevier Science Ltd[ All rights reserved 9859Ð9668:88:,! see front matter PII] S9859!9668"87#99907!5 Origin and Control of Dynamics of Hexagonal Patterns in a Photorefractive Feedback System M[ SCHWAB\$ M[ SEDLATSCHEK\ B[ THU RING\ C[ DENZ and T[ TSCHUDI Institut fur Angewandte Physik\ Technische Universitat Darmstadt\ Hochschulstr[ 5\ D!5378 Darmstadt\ Germany Abstract*We investigate the origin of dynamics of hexagonal pattern formation in a photorefractive single feedback system[ By introducing an asymmetry\ we induced di}erent complex motion behaviours[ A visualization method is presented and means to control the dynamics in the case of _xed external parameters are discussed[ Þ 0888 Elsevier Science Ltd[ All rights reserved[ 0[ TRANSVERSE STRUCTURES IN PHOTOREFRACTIVE MEDIA In a variety of nonlinear optical media\ spontaneous pattern formation due to light beam interaction can be observed[ For collinear pump beams\ the cylindrical symmetry of the con! _guration around the propagation axis gives rise to a predominantly hexagonal structure[ Pre! vious experiments were performed using atomic vapor ð0ł\ liquid crystals ðł\ liquid crystal light valves ð2ł\ bacteriorhodopsinð3ł and photorefractive crystals ð4\ 5Ł[ A linear stability analysis of the coupled di}erential equations for the case of contradirectional two!wave mixing was performed recently ð6ł[ Above a certain threshold\ the light intensity pro_le shows a modulational instability in the transverse direction\ giving rise to spatial sidebands with a certain transverse wave vector K T \ determined by the minimum of the corresponding threshold curve[ A self!organizing process then leads to an equidistant spot array in the far _eld determined by the same K T [ The simplest realization of this condition is a complete hexagonal structure which can be monitored in the far _eld or as a honeycomb structure in the near _eld ðsee Fig[ 0"a\b#Ł[ Here\ we present investigations on the origin and control of hexagonal pattern dynamics arising in a photorefractive single feedback system[ After recalling the theoretical aspects\ we introduce the experimental setup and the essential properties of this system[ A method to visualize the dynamics of hexagonal pattern is presented and di}erent motion behaviours are discussed[ [ THEORETICAL TREATMENT A schematic representation of the studied system is given in Fig[ [ The photorefractive medium is pumped by a frequency!doubled Nd]YAG Laser "l42 nm#[ Using a thin biconvex lens\ a double!pass fðf imaging system with a feedback mirror at its end provides the feedback beam[ $ Author for correspondence[ E!mail] michael[schwabýphysik[tu!darmstadt[de[ 690

2 69 M[ SCHWAB et al[ Fig[ 0[ "a# Honeycomb structure in the near _eld "image of the beam at the end of the crystal#[ "b# Hexagonal structure in the far _eld with second and third harmonics[ Fig[ [ Schematic experimental setup*sbspatial sidebands\ Mmirror\ v[m[virtual mirror\ Lpropagation length[ Using ABCD matrices\ one can show that this system is not completely equivalent to a simple mirror feedback setup\ but it has the advantage that the virtual mirror can be moved even inside the crystal by moving the feedback mirror\ i[e[\ negative propagation lengths can be achieved[ The physical role of the feedback path with a virtual mirror position L from the crystal is to introduce a phase lag of k 9 u L between the pump "or the feedback# and the emerging sidebands emitted at an angle u with respect to the propagation axis[ To analyze the threshold condition for the instability of wave vectors in the transverse plane\ we start with the usual two!wave mixing equations\ taking into account the beam pro_le in the transverse direction[ With the assumption that re~ection gratings are dominant in this con_guration\ these coupling equations can be written as ð6ł F z i =B= 9_Fig k 9 n 9 =F= =B= F "0# B z i =F= 9_B ig k 9 n 9 =F= =B= B "# where F and B are the amplitudes of the forward and backward wave "pump and feedback#\ z is the direction of propagation\ k 9 the wave number in vacuum\ n 9 denotes the linear refractive index of the nonlinear medium\ g is the complex photorefractive coupling coe.cient and 9_ denotes the transverse Laplacian[ The amplitudes of the forward and backward wave with weak modulations of wave vector k _ in the transverse direction can be written as F"r#F 9 "z# ð0 F 0 "z#exp"ik _=r _ # F 0 "z#exp" ik _=r _ #Ł "2# B"r#B 9 "z# ð0 B 0 "z#exp"ik _=r _ # B 0 "z#exp" ik _=r _ #Ł "3#

3 Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system 692 Fig[ 2[ Threshold curve for normalized propagation length n 9 L:l 9[6 "solid line# and curve of the minima for all parameter values 0¾n 9 L:l¾0 "dashed#[ Selected parameter values are n 9 L:l 0 and 9 "point 0#\ n 9 L:l 9[6 and 9[2 "#\ n 9 L:l 9[4 "2# and n 9 L:l0 "3#[ where F 20 and B 20 are the relative amplitudes of the spatial sidebands[ Assuming a feedback re~ectivity of unity and no absorption in the medium "see ð6ł for details#\ the threshold condition for transverse instability can be obtained[ In Fig[ 2\ an exemplary plot of the threshold coupling strength gl is shown "solid line# for the normalized propagation length n 9 L:l 9[6 "virtual mirror inside the crystal with length l#[ The graph of the minima of the threshold curves for 0¾n 9 L:l¾0 is displayed in the same _gure as a dashed line[ This parameter curve starts for n 9 L:l 0 "point 0# and increases via point "n 9 L:l 9[6#[ The turning point is denoted as 2 with a parameter value of n 9 L:l 9[4[ Then\ the direction is reversed passing point at n 9 L:l 9[2\ point 0 at n 9 L:l9 and ending at point 3 at n 9 L:l0[ The corresponding angles of sideband generation as a function of the normalized propagation length are shown in Fig[ 3[ The angle of the sidebands increases with larger L\ transits over a sharp rectangular!like plateau and decreases again\ showing a symmetry around the central region of the crystal[ 2[ EXPERIMENTAL PATTERN OBSERVATION Figure 4 shows the experimental setup in detail[ We used a frequency!doubled Nd]YAG Laser operating at a wavelength of 42 nm with a maximum output power of 099 mw[ The nominally undoped KNbO 2!crystal measured l4[5 mm along the c!axis and was slightly tilted in order to reduce the in~uence of re~ections from its surfaces[ The direction of the crystal s c!axis was chosen to be in the direction of the pump beam\ which leads to depletion of the input and ampli_cation of the feedback beam[ The beam diameter inside the crystal was approximately 249 mm\ the power incident on the crystal was mw and the polarization was chosen to be in the direction of the crystal s a!axis in order to take advantage of the large r 02!component of the electrooptic tensor[ Lens 0\ with a focal length of f099 mm\ was used to focus the beam into the crystal with the beam waist at its back surface[ The fðf!feedback system provides the feedback beam with a controllable position of the feedback mirror\ i[e[\ a change in the transverse scale is possible[

4 693 M[ SCHWAB et al[ Fig[ 3[ Sideband angles u of the far _eld structure as function of the normalized propagation length n 9 L:l\ solid line] theory\ squares] experiment[ Fig[ 4[ Experimental setup*llens\ Mmirror\ BSbeamsplitter\ MLSmicroscope lens system[ Beam splitter 0 enables us to monitor the patterns in the near and far _eld\ and the beam diameter\ simultaneously[ The squares in Fig[ 3 indicate the experimental results for exactly counterpropagating beams\ showing a good agreement with the theoretical treatment[ Near the center of the crystal\ no patterns could be observed in our recent experiments[ This is\ in fact\ the region where Honda et al[ observed squares and squeezed hexagons ð7ł[ Maybe the photorefractive coupling constant g was too small in the present case\ as the dashed curve in Fig[ 2 indicates[ With gl¼5\ a parameter region of 9[6¾n 9 L:l¾ 9[2 exists\ for which the threshold for sideband generation is above

5 Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system 694 Fig[ 5[ Visualization method of Thuring et al[ ð09ł[ the value of gl in the present system[ Thus\ sidebands with angles u max cannot be generated[ With u max 9[8> "see Fig[ 2#\ the estimated photorefractive coupling strength for the present study is gl4[4[ A change of the photorefractive coupling constant\ g\ by changing the input beam polarization and a change of the intensity of the incident beam did not result in a measurable change of the transverse scale[ Asymmetries were induced by shifting lens 0 in the transverse direction[ As a result of these asymmetries\ the photorefractive index grating is read out with a di}erent angle\ causing a ~ow in the near _eld ð8ł[ Thus\ various di}erent spatio!temporal structures can be observed in the far _eld[ At the same time\ the asymmetry is a parameter to control the collinearity of the pump and feedback beams\ i[e[\ static hexagonal structures can be achieved in any parameter region by this method[ If the re~ectivity of the feedback system was reduced\ a remarkable transition could be observed] In the region of a feedback re~ectivity of 9[4³r³9[24\ roll patterns were favoured\ making it possible to observe competition between roll and hexagonal patterns when a small misalignment of the pump with respect to the feedback beam was present[ Exactly collinear beams lead to an hexagonal pattern with emphasis on two spots[ 3[ PATTERN DYNAMICS INDUCED BY NONCOLLINEAR PUMP BEAMS We are interested in the visualization of the dynamics in this system when a small non! collinearity is introduced and use the method proposed in ð09ł[ Figure 5 gives a quick survey on this e}ective visualization method[ The distribution of spots on a circle with radius k 0 is projected to a linear dimension with the polar coordinate f as the cartesian axis[ The time axis is chosen to be perpendicular to this direction and enables us to visualize the dynamics of the system[ As a result\ every movement of the spots on the circle results in an appropriate pattern change in the cartesian coordinates[ For example\ a static hexagon causes six stripes\ travelling parallel in the positive y!direction "time!axis#[ The major advantage of this method is that a single image provides us with all the information on the dynamics instead of a sequence of many single pattern images[ Various types of pattern motion have been observed for a maximum feedback re~ectivity of r9[84[ The favoured situation is the one depicted in Fig[ 6[ Two stable spots "rolls# and a rocking motion ð00ł of the other four can be observed\ causing us to name it {Rock n Roll motion [ The axis determined by the stable spots is always perpendicular to the lens shift and\ as a consequence\ to the periodic ~ow in the near _eld[ The frequency of the motion depends linearly on the magnitude of the introduced beam misalignment[ Even the orientation of rotation can be reversed by changing the direction of misalignment[ In the special case of larger misalignments " 9[2>#\ the movement of the other spots seems to vanish and a roll pattern is left in the far _eld[ Another type of motion is the one depicted in Fig[ 7[ Here\ a slow rotation and a fast leap back

6 695 M[ SCHWAB et al[ Fig[ 6[ Rock n Roll motion with two stable spots "opposite to one another# and a periodic rocking motion of the other four[ Left] Temporal evolution\ middle and right] motion snapshots indicating the stable spots for this motion[ Fig[ 7[ Rocking motion with all six spots rotating and jumping back to their initial position[ Left] Temporal evolution\ middle and right] motion snapshots indicating slow rotation and fast leap back[ to the initial position characterizes this movement\ which is called rocking motion ð00ł[ Even this motion is temporally stable\ but control of the frequency is not as easy as in the case of the Rock n Roll motion[ A small change in the misalignment causes more complex rocking motions\ as indicated in Fig[ 8[ The scenario given here consists of a rocking motion with two di}erent time scales[ The regular rocking motion is evident\ but even the single spots wobble in a periodic manner with a higher frequency than the one of the rocking motion[ The exact parameter regions\ e[g[\ the corresponding angles of misalignment for these di}erent types of motion are currently under investigation[ 4[ CONCLUSIONS We have shown that the dynamics of hexagonal pattern formation in a photorefractive feedback system can be induced and controlled by the noncollinearity of the pump!and feedback beams\

7 Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system 696 Fig[ 8[ Complex rocking motion containing two di}erent time scales[ Left] Temporal evolution\ middle and right] motion snapshots with indicated movement and preferred spots\ the other four spots wobble periodically[ whereas the transverse scale depends only on the propagation length[ A variety of di}erent scenarios can occur\ e[g[\ situations with stable and moving spots named {Rock n Roll motion and more complicated rotation!like motions[ Further investigations concerning the parameter regions for these di}erent types of motion are currently in progress[ Acknowled`ements*The authors acknowledge Dr[ T[ Honda for helpful discussions and for providing the KNbO 2!crystal[ The experimental work was partially supported by the Deutsche Forschungsgemeinschaft\ Sonderforschungsbereich 074\ {Nichtlineare Dynamik [ REFERENCES 0[ Petrossian\ A[\ Pinard\ M[\ Ma(¼tre\ A[\ Courtois\ J[ Y[\ Grynberg\ G[\ Transverse pattern formation for coun! terpropagating beams in rubidium vapor[ Europhys[ Lett[\ 088\ 07\ 578[ [ Tamburrini\ M[\ Bonavita\ M[\ Wabnitz\ S[\ Santamato\ E[\ Hexagonally patterned _lamentation in a thin liquid! crystal _lm with single feedback mirror[ Optics Letters\ 0882\ 07\ 744[ 2[ Thuring\ B[\ Neubecker\ R[\ Tschudi\ T[\ Transverse pattern formation in LCLV feedback system[ Optics Commun\ 0882\ 09\ 000[ 3[ Gluckstad\ J[\ Sa}man\ M[\ Spontaneous pattern formation in a thin _lm of bacteriorhodopsin with mixed absorptive dispersive nonlinearity[ Optics Letters\ 0884\ 9\ 440[ 4[ Honda\ T[\ Hexagonal pattern formation due to counterpropagation in KNbO 2 [ Optics Letters\ 0882\ 07\ 487[ 5[ Honda\ T[\ Matsumoto\ H[\ Buildup of spontaneous hexagonal patterns in photorefractive BaTiO 2 with a feedback mirror[ Optics Letters\ 0884\ 9\ 0644[ 6[ Honda\ T[\ Banerjee\ P[\ Threshold for spontaneous pattern formation in re~ection!grating!dominated pho! torefractive media with mirror feedback[ Optics Letters\ 0885\ 0\ 668[ 7[ Honda\ T[\ Matsumoto\ H[\ Sedlatschek\ M[\ Denz\ C[\ Tschudi\ T[\ Spontaneous formation of hexagons\ squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror[ Optics Commun[\ 0886\ 022\ 82[ 8[ Honda\ T[\ Flow and controlled rotation of spontaneous optical hexagon in KNbO 2 [ Optics Letters\ 0884\ 9\ 740[ 09[ Thuring\ B[\ Schreiber\ A[\ Kreuzer\ M[\ Tschudi\ T[\ Spatio!temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system[ Physica D\ 0885\ 85\ 7[ 00[ Mamaev\ A[ V[\ Sa}man\ M[\ Modulational instability and pattern formation in the _eld of noncollinear pump beams[ Optics Letters\ 0886\ \ 72[